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Bifurcations of relative equilibria near zero momentum in Hamiltonian systems with spherical symmetry
Periodic orbits in the Kepler-Heisenberg problem
1. | Department of Mathematics and Statistics, California State University, Sacramento, 6000 J St., Sacramento, CA, United States |
References:
[1] |
R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin-Cummings, 1978. |
[2] |
A. Albouy, Projective dynamics and classical gravitation, Regul. Chaotic Dyn., 13 (2008), 525-542. arXiv:math-ph/0501026v2.
doi: 10.1134/S156035470806004X. |
[3] |
V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978. |
[4] |
D. Barilari, U. Boscain and R. Neel, Small time heat kernel asymptotics at the sub-Riemannian cut locus, Journal of Differential Geometry, 92 (2012), 373-416. |
[5] |
G. Bliss, The problem of Lagrange in the calculus of variations, American J. Math., 52 (1930), 673-744.
doi: 10.2307/2370714. |
[6] |
O. Bolza, Calculus of Variations, $2^{nd}$ edition, Chelsea, 1960. |
[7] |
R. W. Brockett, Control theory and singular Riemannian geometry, in New Directions in Appl. Math., (eds. P.J. Hilton and G.S. Young), Springer-Verlag, (1982), 11-27. |
[8] |
A. Chenciner and R. Montgomery, A remarkable periodic solution of the three body problem in the case of equal masses, Annals of Math., 152 (2000), 881-901.
doi: 10.2307/2661357. |
[9] |
F. Diacu, E. Perez-Chavela and M. Santoprete, The n-body problem in spaces of constant curvature. Part I: Relative equilibria, Journal of Nonlinear Science, 22 (2012), 247-266.
doi: 10.1007/s00332-011-9116-z. |
[10] |
G. Folland, A fundamental solution for a subelliptic operator, Bulletin of the AMS, 79 (1973), 373-376.
doi: 10.1090/S0002-9904-1973-13171-4. |
[11] |
I. M. Gelfand and S. V. Fomin, Calculus of Variations, Dover, 1963. |
[12] |
W. B. Gordon, A minimizing property of Keplerian orbits, American J. Math., 99 (1977), 961-971.
doi: 10.2307/2373993. |
[13] |
P. Griffiths, Exterior Differential Systems and the Calculus of Variations, Birkhäuser, 1983. |
[14] |
N. I. Lobachevsky, The new foundations of geometry with full theory of parallels, in Collected Works, II (1949), 1835-1838, (Russian), GITTL, Moscow. |
[15] |
R. Montgomery, A Tour of Subriemannian Geometries, AMS, 2002. |
[16] |
R. Montgomery and C. Shanbrom, Keplerian dynamics on the Heisenberg group and elsewhere, to appear in Geometry, Mechanics and Dynamics: The Legacy of Jerry Marsden, Fields Institute Communications series. |
[17] |
R. Palais, The principle of symmetric criticality, Commun. Math. Phys, 69 (1979), 19-30.
doi: 10.1007/BF01941322. |
[18] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, 1962. |
[19] |
C. Shanbrom, Two Problems in Sub-Riemannian Geometry, Ph.D thesis, UC Santa Cruz, 2013. |
[20] |
L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, $2^{nd}$ edition, Chelsea, 1980. |
show all references
References:
[1] |
R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin-Cummings, 1978. |
[2] |
A. Albouy, Projective dynamics and classical gravitation, Regul. Chaotic Dyn., 13 (2008), 525-542. arXiv:math-ph/0501026v2.
doi: 10.1134/S156035470806004X. |
[3] |
V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer-Verlag, 1978. |
[4] |
D. Barilari, U. Boscain and R. Neel, Small time heat kernel asymptotics at the sub-Riemannian cut locus, Journal of Differential Geometry, 92 (2012), 373-416. |
[5] |
G. Bliss, The problem of Lagrange in the calculus of variations, American J. Math., 52 (1930), 673-744.
doi: 10.2307/2370714. |
[6] |
O. Bolza, Calculus of Variations, $2^{nd}$ edition, Chelsea, 1960. |
[7] |
R. W. Brockett, Control theory and singular Riemannian geometry, in New Directions in Appl. Math., (eds. P.J. Hilton and G.S. Young), Springer-Verlag, (1982), 11-27. |
[8] |
A. Chenciner and R. Montgomery, A remarkable periodic solution of the three body problem in the case of equal masses, Annals of Math., 152 (2000), 881-901.
doi: 10.2307/2661357. |
[9] |
F. Diacu, E. Perez-Chavela and M. Santoprete, The n-body problem in spaces of constant curvature. Part I: Relative equilibria, Journal of Nonlinear Science, 22 (2012), 247-266.
doi: 10.1007/s00332-011-9116-z. |
[10] |
G. Folland, A fundamental solution for a subelliptic operator, Bulletin of the AMS, 79 (1973), 373-376.
doi: 10.1090/S0002-9904-1973-13171-4. |
[11] |
I. M. Gelfand and S. V. Fomin, Calculus of Variations, Dover, 1963. |
[12] |
W. B. Gordon, A minimizing property of Keplerian orbits, American J. Math., 99 (1977), 961-971.
doi: 10.2307/2373993. |
[13] |
P. Griffiths, Exterior Differential Systems and the Calculus of Variations, Birkhäuser, 1983. |
[14] |
N. I. Lobachevsky, The new foundations of geometry with full theory of parallels, in Collected Works, II (1949), 1835-1838, (Russian), GITTL, Moscow. |
[15] |
R. Montgomery, A Tour of Subriemannian Geometries, AMS, 2002. |
[16] |
R. Montgomery and C. Shanbrom, Keplerian dynamics on the Heisenberg group and elsewhere, to appear in Geometry, Mechanics and Dynamics: The Legacy of Jerry Marsden, Fields Institute Communications series. |
[17] |
R. Palais, The principle of symmetric criticality, Commun. Math. Phys, 69 (1979), 19-30.
doi: 10.1007/BF01941322. |
[18] |
L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, 1962. |
[19] |
C. Shanbrom, Two Problems in Sub-Riemannian Geometry, Ph.D thesis, UC Santa Cruz, 2013. |
[20] |
L. C. Young, Lectures on the Calculus of Variations and Optimal Control Theory, $2^{nd}$ edition, Chelsea, 1980. |
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